Which of the following statements are true? Justify your answers. (This means that if you
think a statement is false, give a short proof or an example that shows it is false. If it is
true, give a short proof for saying so. For instance, to show that ‘{1, padma, blue} is a
set’ is true, you need to say that this is true because it is a well-defined collection of 3
objects.)
i) For any two sets A and B, A B
c A ∩ = B .
ii) The matrix
0 0
1 1
is singular.
iii) The contrapositive of ‘ ∃ y such that )y(P is true’ is ‘ ∈Z ∃ x such that )x(P ∈Z
is true’.
iv) The system 1 2x − 3y = and 0 6y − 4x + 2 = has a unique solution.
v) If y,x such that y ∈C x
2
= and x y
2
= , then 1 x = y = .
vi) a ≥ b ⇔ −a ≤ −b is an absolute inequality.
vii) If }2 A = ϕ,B = },2,1{ C = {− ,1 − , then A× B×C has 4 elements.
viii) The argument of 1+ i3 is
3
π
.
ix) A linear equation over R can have at most one root in C R .
x) x1 − x2 = x1 − x2 ∀x1 x,
2 ∈R .
